Simple and Fast Algorithm for Binary Integer and Online Linear Programming

Li, Xiaocheng; Sun, Chunlin; Ye, Yinyu

doi:10.48550/arxiv.2003.02513

Cited by 9 publications

(14 citation statements)

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“…Notably, the problem has been studied under either the stochastic input model where the coefficient in the objective function, together with the corresponding column in the constraint matrix is drawn from an unknown distribution P, or the random permutation model where they arrive in a random order. As noted in the paper (Li et al, 2020), the random permutation model exhibits similar concentration behavior as the stochastic input model.…”

Section: Other Related Literaturementioning

confidence: 63%

“…Our algorithm is motivated by the traditional online gradient descent (OGD) algorithm (Hazan, 2016). The OGD algorithm applies a linear update rule according to the gradient information at the current period and has been shown to work well in the stationary setting, even when the distribution is unknown (Lu et al, 2020;Sun et al, 2020;Li et al, 2020). However, the update in OGD only involves historical information and for the non-stationary setting, we have to incorporate the prior estimates of the future time periods.…”

Section: Main Results and Contributionsmentioning

confidence: 99%

“…Returning to the point mentioned earlier on the knowledge of the centric distribution PT , it does not matter we know it or not; because as long as all the prior estimate distributions Pt are the same over time, we always have the same budget allocation plan. One step further, when all the P t 's are the same, which means the variation budget W T = 0, Algorithm 3 and its analysis collapse into several recent studies on the gradient-based online algorithm under a stationary environment (Lu et al, 2020;Li et al, 2020).…”

Section: Algorithm and Regret Analysismentioning

confidence: 99%

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Online Stochastic Optimization with Wasserstein Based Non-stationarity

Jiashuo¹,

Li²,

Zhang³

2020

Preprint

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We consider a general online stochastic optimization problem with multiple budget constraints over a horizon of finite time periods. In each time period, a reward function and multiple cost functions are revealed, and the decision maker needs to specify an action from a convex and compact action set to collect the reward and consume the budget. Each cost function corresponds to the consumption of one budget. In each time period, the reward function and the cost functions are drawn from an unknown distribution, which is non-stationary across time. The objective of the decision maker is to maximize the cumulative reward subject to the budget constraints. This formulation captures a wide range of applications including online linear programming and network revenue management, among others. In this paper, we consider two settings:(i) a data-driven setting where the true distribution is unknown but a prior estimate (possibly inaccurate) is available; (ii) an uninformative setting where the true distribution is completely unknown. We propose a unified Wasserstein-distance based measure to quantify the inaccuracy of the prior estimate in setting (i) and the non-stationarity of the system in setting (ii). We show that the proposed measure leads to a necessary and sufficient condition for the attainability of a sublinear regret in both settings. For setting (i), we propose an informative gradient descent algorithm. The algorithm takes a primal-dual perspective and it integrates the prior information of the underlying distributions into an online gradient descent procedure in the dual space. The algorithm also naturally extends to the uninformative setting (ii). Under both settings, we show the corresponding algorithm achieves a regret of optimal order. In numerical experiments, we demonstrate how the proposed algorithms can be naturally integrated with the re-solving technique to further boost the empirical performance.

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Section: Other Related Literaturementioning

confidence: 63%

Section: Main Results and Contributionsmentioning

confidence: 99%

Section: Algorithm and Regret Analysismentioning

confidence: 99%

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Online Stochastic Optimization with Wasserstein Based Non-stationarity

Jiashuo¹,

Li²,

Zhang³

2020

Preprint

Self Cite

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“…In the setting of stochastic input models with linear reward functions it is possible to guarantee an optimal order of regret of O(T 1/2 ) [3,21]. More recent works by Li et al [34] and Balseiro et al [11] propose simple algorithms with O(T 1/2 ) regret guarantees that, in contrast with previous works, do not require periodically solving large linear programs. Finally, under some specific structural assumptions, it is possible to obtain a regret bound of order O(log T ) [33,32].…”

Section: A Related Workmentioning

confidence: 99%

The Parity Ray Regularizer for Pacing in Auction Markets

Celli¹,

Colini-Baldeschi²,

Kroer³

et al. 2021

Preprint

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Budget-management systems are one of the key components of modern auction markets. Internet advertising platforms typically offer advertisers the possibility to pace the rate at which their budget is depleted, through budget-pacing mechanisms.We focus on multiplicative pacing mechanisms in an online setting in which a bidder is repeatedly confronted with a series of advertising opportunities. After collecting bids, each item is then allocated through a single-item, second-price auction. If there were no budgetary constraints, bidding truthfully would be an optimal choice for the advertiser. However, since their budget is limited, the advertiser may want to shade their bid downwards in order to preserve their budget for future opportunities, and to spread expenditures evenly over time. The literature on online pacing problems mostly focuses on the setting in which the bidder optimizes an additive separable objective, such as the total click-through rate or the revenue of the allocation. In many settings, however, bidders may also care about other objectives which oftentimes are non-separable, and therefore not amenable to traditional online learning techniques. Building on recent work, we study the frequent case in which advertisers seek to reach a certain distribution of impressions over a target population of users. We introduce a novel regularizer to achieve this desideratum, and show how to integrate it into an online mirror descent scheme attaining the optimal order of sub-linear regret compared to the optimal allocation in hindsight when inputs are drawn independently, from an unknown distribution. Moreover, we show that our approach can easily be incorporated in standard existing pacing systems that are not usually built for this objective. The effectiveness of our algorithm in internet advertising applications is confirmed by numerical experiments on real-world data.Preprint. Under review.

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“…[4,6,8,9]. Some recent works study the efficiency on converge [2,3,13], extension versions for more general setting [1,11,12], and more complicated user models [7,17]. Different from online matching, online convex optimization involves learning frameworks and has been studied in theory and practice [10].…”

Section: Related Workmentioning

confidence: 99%

A General Traffic Shaping Protocol in E-Commerce

Shen¹,

Huzhang²,

Zhou³

et al. 2021

Preprint

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To approach different business objectives, online traffic shaping algorithms aim at improving exposures of a target set of items, such as boosting the growth of new commodities. Generally, these algorithms assume that the utility of each user-item pair can be accessed via a well-trained conversion rate prediction model. However, for real E-Commerce platforms, there are unavoidable factors preventing us from learning such an accurate model. In order to break the heavy dependence on accurate inputs of the utility, we propose a general online traffic shaping protocol for online E-Commerce applications. In our framework, we approximate the function mapping the bonus scores, which generally are the only method to influence the ranking result in the traffic shaping problem, to the numbers of exposures and purchases. Concretely, we approximate the above function by a class of the piece-wise linear function constructed on the convex hull of the explored data points. Moreover, we reformulate the online traffic shaping problem as linear programming where these piece-wise linear functions are embedded into both the objective and constraints. Our algorithm can straightforwardly optimize the linear programming in the prime space, and its solution can be simply applied by a stochastic strategy to fulfill the optimized objective and the constraints in expectation. Finally, the online A/B test shows our proposed algorithm steadily outperforms the previous industrial level traffic shaping algorithm. CCS CONCEPTS• Theory of computation → Theory and algorithms for application domains; • Applied computing → Online shopping.

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Simple and Fast Algorithm for Binary Integer and Online Linear Programming

Cited by 9 publications

References 0 publications

Online Stochastic Optimization with Wasserstein Based Non-stationarity

Online Stochastic Optimization with Wasserstein Based Non-stationarity

The Parity Ray Regularizer for Pacing in Auction Markets

A General Traffic Shaping Protocol in E-Commerce

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